Hexagonal Quasicrystals Research Articles

Impact of frictional heat and electric charge on the thermomechanical contact properties of piezoelectric quasicrystals: A DC-FFT algorithm

Piezoelectric quasicrystals (PEQCs) are quasicrystals with piezoelectricity. It combines the excellent properties of piezoelectric materials and quasicrystal materials, and is expected to be used as actuators in fields such as aerospace, automotive, and intelligent manufacturing. This article investigates the physical and mechanical properties of one-dimensional hexagonal piezoelectric quasicrystals (1D HPEQCs) coating under normal force and frictional heat and current density by using the discrete convolution-fast Fourier transform (DC-FFT) algorithm-based hybrid element method (HEM). The general solution of the model in the two-dimensional Fourier domain is obtained by combining the two-dimensional Fourier transform with strict operator theory and the Almansi’s theorem. The DC-FFT algorithm-based HEM is used to efficiently and accurately cope with the issue of transforming the fundamental solution in the Fourier domain into the spatial domain. Therefore, the numerical solutions and distribution of stress and displacement in the phonon and phason fields are obtained. The results indicate that current density has a significant impact on the phason field, and an increase in current density enhances the mechanical behavior of the phason displacement. The research results can provide theoretical guidance for 1D HPEQCs as intelligent coating materials.

Thermoelastic fracture of two-dimensional hexagonal quasicrystal media weakened by a penny-shaped crack subjected to uniformly antisymmetric heat fluxes

This study focuses on the thermoelastic fracture behavior of quasicrystals, a class of solids that exhibit unique properties between traditional crystals and amorphous materials. The complex atomic structure of quasicrystals poses challenges in understanding their fracture mechanisms under thermoelastic loading conditions. Central to our investigation is an analytical examination of a penny-shaped crack in an infinite three-dimensional body composed of two-dimensional hexagonal quasicrystals, where the upper and lower crack surfaces are applied to a pair of uniformly antisymmetric heat fluxes. This crack problem requires simultaneous consideration of thermal-phason-phonon multiple field coupling. According to the symmetry of field variables with respect to the crack plane, the thermal-phason-phonon coupled crack problem is transformed into a mixed boundary value problem in the upper half space. The extended displacement discontinuities, encompassing both phonon and phason displacement discontinuities, as well as the temperature discontinuity, are chosen as the basic unknown variables to construct the boundary integral-differential equations governing the mixed boundary value problem. Based on these boundary governing equations and Fabrikant's potential theory method, the problem with the crack surface subjected to uniform antisymmetric heat fluxes is solved. The solutions of thermal-phason-phonon fields on the crack plane, and in the full space are given in closed-form. Numerical results are employed to validate the obtained analytical solutions and visually illustrate the spatial distribution of thermal-phonon-phason coupling fields in the vicinity of the crack. The study provides fundamental insights into the behavior of cracks in quasicrystals under thermal loading, with potential implications for the design of new materials and structures.

Analytical solutions to Mode I penny-shaped crack problems in two-dimensional hexagonal quasicrystals with piezoelectric effect

The paper studies a penny-shaped crack in an infinite three-dimensional body of two-dimensional hexagonal quasicrystal media with piezoelectric effect. The crack surfaces are applied combined electric and normal phonon loadings. Such a Model I crack problem is transformed into a mixed boundary value problem in the upper half-space, which is analytically solved using Fabrikant's potential theory method. The boundary integral-differential equations governing Model I crack problems are presented for two-dimensional hexagonal piezoelectric quasicrystals. The normal phonon displacement discontinuity and electric potential discontinuity across crack surfaces are taken as the unknown variables of boundary governing equations. Analytical solutions of all field variables are derived not only for the crack plane but also for the full space. Solutions in integral form are provided for the penny-shaped crack under arbitrarily distributed electric and normal phonon loadings. Closed-form solutions in terms of elementary functions are given for concentrated point loadings and uniformly distributed loadings, respectively. Key fracture mechanics parameters, such as crack surface extended displacements (i.e., normal phonon displacement, electric potential), crack tip extended stresses (i.e., normal phonon stress, electric displacement) distribution, and corresponding extended stress intensity factors, are clearly derived. Numerical results are utilized to verify the present analytical solutions and graphically illustrate the distribution of phonon-phason-electric coupling fields around the crack. The present solution can serve as a benchmark for both experimental and numerical investigations.

On the Thermally Induced Interfacial Behavior of Thin Two-Dimensional Hexagonal Quasicrystal Films with an Adhesive Layer

The mechanical response of a quasicrystal thin film is strongly affected by an adhesive layer along the interface. In this paper, a theoretical model is proposed to study a thin two-dimensional hexagonal quasicrystal film attached to a half-plane substrate with an adhesive layer, which undergoes a thermally induced deformation. A perfect non-slipping contact condition is assumed at the interface by adopting the membrane assumption. An analytical solution to the problem is obtained by constructing governing integral–differential equations for both single and multiple films in terms of interfacial shear stresses that are reduced to a linear algebraic system via the series expansion of Chebyshev polynomials. The solution is compared to that without adhesive layers, and the effects of the aspect ratio of films, material mismatch, and the adhesive layer, as well as the interaction between films, are discussed in detail. It is found that the adhesive layer can soften the localized stress concentration. This study is instructive to the accurate safety assessment and functional design of a quasicrystal film system.

Three-Dimensional General Solutions of Orthorhombic Quasicrystals With Constraints

Abstract This study employs the Lur'e operator method to derive generalized solutions for orthorhombic quasicrystals, incorporating anisotropy factors as constraints. The solutions derived contain the Lekhnitskii–Hu–Nowacki and Elliott–Lodge solutions as special cases. The corresponding fundamental solutions or Green's functions within the infinite space are also derived, offering a comprehensive characterization of quasicrystal anisotropy. It is noteworthy that Green's functions in orthorhombic quasicrystals can be simplified to those in hexagonal quasicrystals or conventional orthorhombic crystals, with possible broad engineering applications.

Ising sSon Frustrated Bronze‐Mean Hexagonal Quasicrystal

AbstractWe investigate the Ising model on the Bronze‐mean hexagonal quasicrystal (BMH QC), an aperiodic tiling with geometric frustration. Our extensive Monte Carlo simulations explore the model's rich phase diagram, revealing six distinct phases with diverse magnetic properties and degrees of frustration. We uncover exotic spin glass phases, signaled by the replica symmetry breaking and slow relaxation dynamics. We shed light on the intriguing magnetic properties of frustrated quasicrystals and open new avenues for studying exotic phases in condensed matter physics.

On the Constitutive Modelling of Piezoelectric Quasicrystals

Quasicrystals endowed with piezoelectric properties belong nowadays to novel piezoelectric materials. In this work, the basic framework of generalized piezoelectricity theory of quasicrystals is investigated by providing an improvement of the existing constitutive modelling. It is shown, for the first time, that the tensor of phason piezoelectric moduli is fully asymmetric without any major or minor symmetry, which has important consequences on the constitutive relations as well as on its classification with respect to the crystal systems and Laue classes. The exploration of the tensor of phason piezoelectric moduli has a significant impact on the understanding of the piezoelectric properties of quasicrystals. Using the group representation theory, the classification of the tensor of phason piezoelectric moduli with respect to the crystal systems and Laue classes is given for one-dimensional quasicrystals. The number of independent components of the phason piezoelectric moduli is determined for all 31 point groups of one-dimensional quasicrystals. It is proven that the 10 centrosymmetric crystallographic point groups have no piezoelectric effects and that the remaining 21 non-centrosymmetric crystallographic point groups exhibit piezoelectric effects due to both phonon and phason fields. Moreover, the constitutive relations for one-dimensional hexagonal piezoelectric quasicrystals of Laue class 9 with point group 6 and Laue class 10 with point group 6mm are explicitly derived, showing that the constitutive relations for piezoelectric quasicrystals depend on the considered Laue class as well as on the point group. Comparisons with existing results in the literature and discussion are also given.

Elliptic crack problem under shear mode in one-dimensional hexagonal quasicrystals with crack surface parallel to the quasiperiodic axis

In this work we are concerned with a special elliptic crack problem in one-dimensional hexagonal quasicrystals. The crack surface is assumed to be parallel to the quasiperiodic axis. A couple of anti-symmetrical uniform shear loadings is applied on the crack surfaces. Taking advantage of the potential theory method, we develop the governing integral equation and obtain the phonon–phason field in terms of simple integrals. The crack slip displacement and the stress intensity factor are derived explicitly. Based on the analytical solution, we investigate the effects of phason field, crack orientation and the eccentricity of elliptic crack on the important fracture parameters. The solutions obtained in this work are helpful to understand the fracture behavior of one-dimensional hexagonal quasicrystals, especially for the situations in which the cracks are not located on the isotropic planes.

Analytical solutions for the plane thermoelastic problem of a nano-open crack in one-dimensional hexagonal quasicrystal non-periodic plane

As a stable material at high temperatures, the presence of cracks and other defects greatly reduces the performance of quasicrystals (QCs). The plane thermoelastic problem of one-dimensional hexagonal quasicrystals (1DHQ) non-periodic plane containing a nano-open crack is analyzed using surface elasticity theory. The closed form solutions are obtained to describe the thermoelastic field corresponding to the nano-crack by the Fourier integral transformation technique. The analytical solutions of the thermal stress intensity factors (TSIFs) and the strain energy density factor (SEDF) at the crack tip are derived explicitly. The influence of the applied loads, the surface effects and the multi-field coupling effects on the TSIFs and SEDF are discussed. The numerical results reveal that the surface effects have an obvious impact on TSIFs with small applied loads, and the surface effects will inhibit the propagation of nano-open cracks. The research results of this study provide some reliable theoretical references for the development of nano-QCs.

Interaction of anti‐plane shear waves with two collinear cracks in 1D hexagonal piezoelectric quasicrystals

AbstractThe present article presents the interaction of anti‐plane shear waves by two collinear cracks in one dimensional (1D) hexagonal piezoelectric quasicrystals (PQCs). With the aid of Fourier transform, the mixed boundary value problem (MBVP) is transformed into three pairs of dual integral equations, which are solved analytically by Hilbert transform. The explicit expressions for dynamic stress intensity factors (DSIFs) of phonon and phason fields, crack opening displacement (COD) and electric displacement (ED) are derived in closed form and some special cases are studied. Numerical values of DSIFs of phonon and phason fields, COD and ED are plotted to show the effect of crack length and electric boundary condition. Moreover, the DSIFs of phonon field and COD are represented graphically for single crack.

Analytical Solution of the Interference between Elliptical Inclusion and Screw Dislocation in One-Dimensional Hexagonal Piezoelectric Quasicrystal

This study examines the interference problem between screw dislocation and elliptical inclusion in one-dimensional hexagonal piezoelectric quasicrystals. The general solutions are obtained using the complex variable function method and the conformal transformation technique. When the screw dislocation is located outside or inside the elliptical inclusion, the perturbation method and Laurent series expansion are employed to derive explicit analytical expressions for the complex potentials in the elliptical inclusion and the matrix, respectively. Considering four types of far-field force and electric loading conditions, analytical solutions for various specific cases are obtained by using matrix operations. Expressions for the phonon field stress, phason field stress, and electric displacement are given for special cases, including the absence of a dislocation, the presence of an elliptical hole, and the interference between a screw dislocation and circular inclusion, as well as the case of a circular hole. The design and analysis of quasicrystal inclusion structures can benefit from the results of this work.

An analytic solution of an arbitrary location through-crack emanating from a nano-circular hole in one-dimensional hexagonal piezoelectric quasicrystals

The mode III fracture performance of an arbitrary location through-crack at the edge of a nano-circular hole in one-dimensional hexagonal quasicrystals is studied. Based on the G-M elasticity theory, the piezoelectric quasicrystal theory and the boundary value problems theory, the phonon, phason and electric fields of the nano-defects (nano-hole and nano-cracks) are obtained, and the analytic expressions of the field intensity factors at both ends of through-crack are present. The present solution can be degenerated into the existing results. The relationship between the field intensity factors with the nano-defects related parameters, far-field loads and piezoelectric quasicrystal coupling coefficient are discussed. The field intensity factors have obvious size effects when the defects size is at nanoscale and the surface effect is considered. The field intensity factors tend to be classical theory with the increase in defect size. The field intensity factors show different changes with the increase in crack location angle and relative size between cracks. The size effect of the field intensity factors is significantly affected by the crack location. The far-field mechanical-electric loads and the piezoelectric quasicrystal coupling coefficient have obvious influence on the field intensity factors.

A phase-field model for thermo-elastic fracture in quasicrystals

In this paper, a thermo-elastic phase-field model for brittle fracture in quasicrystals is developed. The kind of quasicrystals is not preset, so this model is suitable for almost all thermo-elastic quasicrystal solids. The phonon force and thermal load trigger the variation of the phonon elastic energy, and then drive crack initiation and propagation. The thermal conductivity in the cracked region is assumed to be degraded, which means cracks are adiabatic. For the static penny-shaped crack problem, the results based on the present model agree well with the existing analytical solution. For the quasi-static crack problems, both the tensile/shear fracture in a thermal environment and the thermal shock cracking failure can be dealt with by the present model. Several examples of 1D hexagonal and 2D decagonal quasicrystals are performed to evaluate the effects of thermal load and phason elastic field. The simulation results show that the thermal load can influence the peak force and fracture displacement. For the single-edge notched tension test, the effect of the thermal load on the peak force is small, but it on the fracture displacement is obvious. For the single-edge notched shear test, the thermal load has an obvious influence on both the peak force and fracture displacement. For the thermal shock cracking problem, the phason elastic field has a significant effect on the cracking time and crack number. The present model can serve as a powerful tool to simulate various thermo-elastic fracture problems in quasicrystals.

A phase-field framework for brittle fracture in quasi-crystals

Most of quasi-crystals are brittle at room temperature due to their specific cluster structures. Phase-field fracture models have demonstrated a powerful ability to predict brittle crack evolution. This paper develops a phase-field framework for modeling macroscopic brittle fracture in quasi-crystal solids. The kind of quasi-crystals is not preset in the phase-field model; therefore, this model is valid for almost all quasi-crystal solids. The phase-field model for the first time introduces a volume fraction parameter into the fracture toughness to reflect the effect of phason wall. Furthermore, a new numerical implementation approach of phase-field fracture models in Comsol Multiphysics is presented. Several examples of 1D hexagonal and 2D decagonal quasi-crystals are performed. The developed model is validated against the reported results of benchmark problems. Further investigations indicate that the phason field peculiar to quasi-crystals has a certain influence on the crack paths in asymmetric fracture problems and on the force–displacement curves in both symmetric and asymmetric fracture problems. When the fracture toughness is fixed, the increase of the volume fraction parameter results in the decrease of the peak force and the increase of the failure displacement. When the influence of the phason field on the fracture toughness is considered, both the peak force and failure displacement would sharply decrease, which indicates that the degradation of fracture toughness due to the phason field plays a more important role than the phason elastic energy in the fracture of quasi-crystals. Furthermore, the relation between the phonon-phason energy transformation and the elastic constants of QCs has been qualitatively analyzed. The developed phase-field modeling framework provides a guidance for understanding the fracture mechanism of quasi-crystals and assessing the safety of quasi-crystal structures in engineering practice.

Stress singularity of one-dimensional hexagonal piezoelectric quasicrystal composites due to thermal effect

In the framework of thermo-electro-elasticity, the present paper investigates the singular behaviors of interface corners, interface cracks, composite wedges and spaces for one-dimensional hexagonal quasicrystal. The stress function and temperature variation can be described as the exponential form with a view to stress and heat flux singularities. Based on the Stroh formalism, the analytical expressions of singular orders of stress and heat flux are easily established by simple multiplication of the crucial matrix. Numerical examples of the singular orders are given for some general cases including single, bi-material, and tri-material wedges and spaces under different boundary conditions. Numerical results show that the geometry structures, material properties, boundary conditions, and heat conduction coefficients have great influences on singularities, but thermal moduli have no effect on singularities.

Study on effective electroelastic properties of one-dimensional hexagonal piezoelectric quasicrystal containing randomly oriented inclusions

In this paper, the effective electroelastic properties of one-dimensional (1D) hexagonal piezoelectric quasicrystal containing randomly oriented inclusions are considered. The explicit expressions are obtained for the Eshelby tensors for 1D hexagonal piezoelectric quasicrystals containing rod-shaped and penny-shaped inclusions. The closed forms of the electroelastic constants are acquired for four special cases of random orientations of inclusions. Numerical results are given for the 1D hexagonal piezoelectric quasicrystal containing randomly oriented ellipsoidal inclusions. The results indicate that the effective electroelastic properties of 1D hexagonal piezoelectric quasicrystal composites are strongly affected by both the aspect ratio and the orientation of inclusions.

Analytical Solution for a 1D Hexagonal Quasicrystal Strip with Two Collinear Mode-III Cracks Perpendicular to the Strip Boundaries

We considered the problem of determining the singular elastic fields in a one-dimensional (1D) hexagonal quasicrystal strip containing two collinear cracks perpendicular to the strip boundaries under antiplane shear loading. The Fourier series method was used to reduce the boundary value problem to triple series equations, then to singular integral equations with Cauchy kernel. The analytical solutions are in a closed form for the stress field, and the stress intensity factors and the energy release rates of the phonon and phason fields near the crack tip are expressed using the first and third complete elliptic integrals. The effects of the geometrical parameters of the crack configuration on the dimensionless stress intensity factors are presented graphically. The studied crack model can be used to solve the problems of a periodic array of two collinear cracks of equal length in a 1D hexagonal quasicrystal strip and an eccentric crack in a 1D hexagonal quasicrystal strip. The propagation of cracks produced during their manufacturing process may result in the premature failure of quasicrystalline materials. Therefore, it is very important to study the crack problem of quasicrystalline materials with defects as mentioned above. It can provide a theoretical basis for the application of quasicrystalline materials containing the above defects.

Dynamic response of a one‐dimensional hexagonal quasicrystal rod in the framework of fractional‐order thermoelasticity

AbstractBased on the fractional‐order thermoelastic theory, the dynamic response of a finite quasicrystal (QC) rod fixed at both ends and subjected to a moving heat source is analyzed. The governing equations of one‐dimensional (1D) hexagonal QC rod are derived. And then the numerical solution is obtained by using the Laplace transform and its numerical inversion. The effects of time; the velocity of heat source; and fractional‐order parameter on displacement, stress, and temperature of the QCs are discussed and compared with classical materials in this paper.

Dynamic fracture of a partially permeable crack in a functionally graded one-dimensional hexagonal piezoelectric quasicrystal under a time-harmonic elastic SH-wave

The dynamic fracture of a crack in a functionally graded one-dimensional (1D) hexagonal piezoelectric quasicrystals (PEQCs) subjected to a time-harmonic elastic SH-wave is studied by using integral transform technique and Copson method. It is assumed that the crack surface is a partially permeable boundary condition and the material properties vary continuously as an exponential function. With the help of the Fourier transform, the boundary value problem of partial differential equation describing fracture problem is formulated to three pairs of dual integral equations, which are numerically solved by Copson method. Explicit expressions for the electroelastic field including phonon and phason stresses and electric field on the crack face are determined. The dynamic intensity factors of the electroelastic field are obtained in closed form, and some special cases of obtained results are discussed. On the basis of theoretical analysis and numerical simulation of the established models, numerical analysis was then conducted to discuss crack length, gradient parameter, electric boundary condition, electric loading, incident angle, amplitude, and wave number on the fracture characteristics of material. The research of this paper will provide a theoretical basis for nondestructive testing, optimal design, and reliability analysis of materials and will enrich the research content of fracture mechanics of multi-field coupling materials.

On energy release rate for an electrically permeable interface crack between two different 1D hexagonal piezoelectric QCs

A bi-material composed of two 1D piezoelectric hexagonal quasicrystals with a crack along the material interface was considered. Phonon and phason remote loading providing plane strain conditions were applied at infinity and an open crack model was adopted. Because electromechanical fields have an oscillating singularity at the crack tip, the energy release rate (ERR) is the most important fracture parameter in this case. The main purpose of this study is to define ERR in an analytical form. Using the obtained earlier asymptotic presentations of the phonon and phason fields at the crack tip and the crack closure integral approach, the desired formula is obtained in a simple form. The eligibility of the formula is verified by means of a crack in the bi-material composed of two 1D piezoelectric hexagonal quasicrystals and by considering a crack in homogeneous materials of this type. An additional comparison of the results obtained for a particular case of an isotropic material with known values confirmed the validity of the obtained formula.